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[Lapack] eigenvector sign question

How do the LAPACK algorithms determine the sign to place on an 

I have experienced a situation in which two similar covariance matrices 
(R1 and R2) have eigen decompositions LDL^T where L = [l_1, l_2, ..., 
l_n] and D is a diagonal matrix of eignevalues.  The respective matrices 
of eigenvectors, L1 and L2, are very similar but the ith eigenvectors 
from R1 and R2 are of different sign.  This doesn't happen all the time 
either; meaning that if I take 40 pairs of covariance matrices, only 3 
pairs will have this sign problem.  Why is this?

- Erich Hefner
Colorado State University

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